ramakanth 100922003
TRANSCRIPT
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SELF EXCITED VIBRATIONS
Mechanical vibrations Seminar on
Ramakanth P Joshi
100922003
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OBJECTIVE
The theory of self excited vibration has important application in mechanicalsystems.
Shortly after the official opening of the Tacomas narrows bridge at pugetsound,washigton, the main span of bridge underwent self excitation oscillationwhich resulted in the destruction of bridge.
steady transversely blowing wind was held responsible
FOCUS ON
Definition: self excited vibration
Mathematical representationDynamic stability analysis for self excited vibration
Friction induced vibration
Flow induced vibration
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DEFINITION: SELF EXCITED VIBRATION
There are systems for which the exciting force is a function of the motion parametersof the system, such as displacement, velocity, or acceleration . Such systems are
called self-excited vibrating systems since the motion itself produces the exciting
force.
In self excited vibration the alternating force that sustains the motion is created or
controlled by the motion itself; when the motion stops the alternating force
disappears,
In a forced vibration the sustaining alternating force exists independently of the
motion and persists even when the vibratory motion is stopped
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MATHEMATICAL REPRESENTATIONself-excited vibration can be considered as a free vibration with negative damping.
Solution for this equation can be written as
Which is clearly a vibration with exponentially increasing amplitude
A mechanical system is statically stable if a displacement from the equilibrium position sets
up a force (or couple) tending to drive the system back to the equilibrium position. It is
statically unstable if the force thus set up tends to increase the displacement. Therefore
static instability means a negative spring constant k
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DYNAMIC STABILITY ANALYSIS FOR SELF EXCITED VIBRATION
A system is dynamically stable if the motion (or displacement)converges or
remains steady with time. On the other hand, if the amplitude of displacement
increases continuously (diverges) with time, it is said to be dynamically
unstable. The motion diverges and the system becomes unstable if energy is fed
into the system through self excitation. To see the circumstances that lead to
instability, we consider the equation of a
single degree of freedom system:
If a solution of the formx(t) = Cest, where C is a constant and Ieads to the characteristicequation
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The roots of the equations are
Roots of equation x(t) = Cest
the motion will be diverging and a periodic if the roots S and S are real and positive.
This situation can be avoided if c/m and k/m are positive.
The motion will also diverge if the roots S and S are complex conjugates with positivereal parts. To analyze the situation, let the roots S and S of Eq. be expressed as
The above equation shows that for negative p, c/m should be positive and for positive p2+q2
,K/m should be positive.
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FRICTION INDUCED VIBRATIONS AND ITS FREQUENCY
Small amplitude behavior
The equation of motion of the system is
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Since the base is moving at constant velocity v1, and the
corresponding coefficient of friction at that velocity is 1,
the system reaches static equilibrium with:
If now a disturbance applies a small positive increment of
velocity, , to the mass, as shown in Fig
But, from the fig
On substitution
Upon subtracting
We now see that gives the change in force for a unit
change in velocity, and is therefore equivalent to a
viscous damping coefficient
It is usually more convenient to write
in the non-dimensional formWhere,
1
2
3
4
5
6
7
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The solution for the equation can be written as
If then the total non-dimensional damping coefficient, , is negative, it can be seen
from the above equation that the exponent becomes positive, and x will always consist
of an exponentially growing oscillation, until limited in some way.
Large amplitude behavior
The amplitude will limit if the energy derived from the friction force over a complete cycle
becomes equal to the energy dissipated by the damper.
Limiting will always occur when the peak velocity due to the oscillation, , exceeds the
mean rubbing velocity, v1
Assuming that the waveform remains sinusoidal, this corresponds to a displacement limit,
xmax, given by:
CONTINUEDu..
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FRICTION-INDUCED VIBRATION IN AIRCRAFT LANDING GEAR
The conventional cantilever type of aircraft landing gear, when fitted with brakes, canexhibit two main forms of friction-induced vibration:
1) When the brakes are working normally, if the coefficient of friction between therubbing surfaces in the brakes tends to decrease as the velocity increases, negative
damping may result, producing the vibration known as brake judder or brakechatter, at low freq and break squeal at high frequencies
2) When the brakes lock completely, perhaps because the anti-skid device, if fitted,does not always operate down to very low speeds, the friction characteristicsbetween the tires and the runway can then provide the negative damping instead.
Measurements of tire friction show that the negative slope of coefficient of frictionversus v, necessary for instability, tends to occur when the runway is wet, and thespeed is low.
In both the cases it will lead to fatigue failure
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DYNAMIC INSTABILITY CAUSED BY FLUID FLOWH
igh-tension electric transmission lines have been observed under certain weatherconditions to vibrate with great amplitudes and at a very slow frequency.
A rough calculation shows that the natural frequency of the span is of the same order
as the observed frequency.
We have a case of self-excited vibration caused by the wind acting on the wire which,
on account of the accumulated sleet, has taken a noncircular cross section
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STABILITY CRITERIA
In aerodynamic work it is customary to resolve the total air force on an object into twocomponents:
a. In the direction of the wind (the drag or resistance D).
b. Perpendicular to the wind (the lift L).
These two forces can be measured easily with the standard windtunnel apparatus.
= tan-1(v/V)
The lift and drag forces Land D have vertical upward component (i.e., components
opposite to the direction of the motion) of Lcos and D sin . The total upward damping
force F of the wind is
The criterion for dynamic stability is
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CONTINUED..
Thus the system will be unstable when
The values of the lift and drag of an arbitrary cross section cannot be calculated from
theory but can be found from a wind-tunnel test. The results of such tests are usually
plotted in the form of a diagram such as
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KARMAN VORTICES
When a fluid flows by a cylindrical obstacle, the wake behind the obstacle is no
longer regular but in it will be found distinct vortices of the pattern shown in Fig. This phenomenon has been studied experimentally
The cylinder moves forward by about 41/2 diameters during one period of the
vibration.
The eddy shedding on alternate sides of the cylinder causes a harmonically varying
force on the cylinder in a direction perpendicular to that of the 'stream.
The maximum intensity of this force can be written in the form usual for most
aerodynamic forces (such as lift and drag) as follows
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REDUCTION OF FLOW INDUCED VIBRATION
In high speed cars the flow induced lift
forces can unload the tires thereby causing
problems with steering control and stabilityof the vehicle. Although lift forces can be
countered partly by adding spoilers ,the
drag forces will increase. In recent years
movable inverted airfoils are being used to
develop a downward aerodynamic force
with improved stability characterstics
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REFERENCES
Mechanical vibrations, J.P. DEN HARTOG, Fourth edition, McGraw-Hill book company
Mechanical vibrations,Singiresu S Rao,fourth edition, ISBN 81 7758-874-
5,Pearson sducation.
Shock and vibration handbook,Cyril M. Harris and Charles E. Crede, volume
1,McGRaw hill company.